I am a strawberry blonde myself and I have dyed my hair brown before and it looked good you just got to make sure to use the permanent hair dye if you want it to not fade cause my hair faded within like 3 weeks :( But yeah also just don't make it too dark
If you are a genuine strawberry blonde I would be inclined to stick with it. You can always go and try some wigs on to see how you would look with brown hair. There are also sites you can upload your picture and try different colours and hairstyles.